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Its kind of boring to do it in a univariate polynomial ring. This is what you want:

sage: R.<x,y,z> = PolynomialRing(GF(2*4, 'a'))
sage: I = R.ideal(x^2+y^2+z^2-4, x^2+2*y^2-5, x*z-1)
sage: I.vector_space_dimension()
4
sage: I.normal_basis()
[y*z, z, y, 1]